Chapter 7 - Techniques of Integration - 7.5 Strategy for Integration - 7.5 Exercises - Page 548: 56

$$\tan^{-1}\sqrt{x}+C$$

Given \begin{aligned} \int \frac{d x}{\sqrt{x}+x \sqrt{x}} \end{aligned} Let $$u^2 =x\ \ \ \to \ \ 2udu=dx$$ then \begin{aligned} \int \frac{d x}{\sqrt{x}+x \sqrt{x}}&=\int \frac{2ud u}{\sqrt{u^2}+u^2 \sqrt{u^2}}\\ &= \int \frac{du}{1+u^2}\\ &=\tan^{-1}u+C\\ &=\tan^{-1}\sqrt{x}+C \end{aligned}